Stackelberg differential game for insurance under model ambiguity

Abstract

We study a dynamic Stackelberg differential game between a buyer and a seller of insurance policies in a spectrally negative Lévy framework, in which both parties are ambiguous about the intensity and severity of insurable losses. Both the buyer and seller aim to maximize their expected wealth, plus a penalty term that reflects ambiguity, over an exogenous random horizon. Under a mean-variance premium principle and a quadratic penalty for ambiguity, we obtain the equilibrium in closed form. Our main results show that the buyer's robust optimal indemnity is a coinsurance with proportion less than one-half, which increases (resp. decreases) as the buyer (resp. seller) becomes more ambiguity averse. Also we show that the seller's robust optimal premium rule equals the net premium under the buyer's optimally distorted probability, which is the buyer's “best hope,” and it exceeds the actuarially fair premium under the seller's optimally distorted probability measure so is, thereby, acceptable to the seller.